Answer each question with a minimum of 70 words per question: Compare and contrast adaptive...

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Answer each question with a minimum of 70 words per question: 	Compare and contrast adaptive quadrature and Gaussian quadrature     	Discuss the pros and cons of the Newton-Cotes formulae and Romberg Integration. Provide specific examples and use scholarly sources to support your points. 
	
Document Preview: Answer each question with a minimum of 70 words per question:Compare and contrast adaptive quadrature and Gaussian quadratureDiscuss the pros and cons of the Newton-Cotes formulae and Romberg Integration. Provide specific examples and use scholarly sources to support your points.

Robert · Accepted Answer

Adaptive quadrature : 
1)
	Adaptive quadrature
	Gaussian quadrature
	Definition :
In applied mathematics, adaptive quadrature is a process in which the integral of a function f(x) is approximated using static quadrature rules on adaptively refined subintervals of the integration domain. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well-behaved" integrands, but are also effective for "badly-behaved" integrands for which traditional algorithms fail.
	Definition :
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi​  and weights wi for i =1,2,...n
	Quadrature rules :
The quadrature rules generally have the form
Qn = 
å
=
n
i
xi
wif
0
)
(
 ~  
ò
b
a
dx
x
f
)
(
Where the nodes xi​ and weights wi are generally pre-computed.
	Quadrature rules : 
The domain of integration for such a rule is conventionally taken as [−1, 1],

Answer each question with a minimum of 70 words per question: Compare and contrast adaptive quadrature and Gaussian quadrature Discuss the pros and cons of the Newton-Cotes formulae and Romberg...

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